In this work we investigate the interaction of charge carriers in graphene with a series of p-n-p junctions arranged according to a deterministic quasiperiodic substitutional Fibonacci sequence. The junctions create a potential landscape with quantum wells and barriers of different widths, allowing the existence of quasi-confined states. Spectra of quasi-confined states are calculated for several generations of the Fibonacci sequence as a function of the wavevector component parallel to the barrier interfaces. The results show that, as the Fibonacci generation is increased, the dispersion branches form energy bands distributed as a Cantor-like set. Besides, for a quasiperiodic set of potential barriers, we obtain the electronic tunneling probability as a function of energy, which shows a striking self-similar behavior for different generation numbers.